Implicitly defined quantities that depend on unknown parameters introduce challenges when they are involved in statistical models. Examples of such quantities are solutions to
parameterized algebraic equations, optimization problems, integrals, or ordinary differential equations (ODEs). They generally do not have a closed form given the parameters, and to evaluate the model likelihood they have to be approximated using numerical methods [1]}. In the Bayesian context, model inference is commonly done by sampling the posterior distribution of the parameters using Markov chain Monte Carlo (MCMC) [2]} techniques. The theory behind MCMC assumes that the model likelihood can be computed exactly, but this is not the case if numerical approximations are required. Some numerical routines can estimate their own error,
but as the true error is not known and its magnitude varies in different parts of the parameter space, it is difficult to predict how it affects the MCMC posterior draws. As this bias has not had a lot of attention in the literature, unaware users can blindly use default configurations of the numerical methods implemented in software packages.
